Question

If $\overrightarrow{u}$ and $\overrightarrow{v}$ are unit vectors and $\theta$ is the acute angle between them,then $2\overrightarrow{u}\times 3\overrightarrow{v}$ is a unit vector for

• A. more than two values of $\theta$
• B. no values of $\theta$
• C. exactly one value of $\theta$
• D. exactly two values of $\theta$

Answer 1

Answer: (C)

exactly one value of $\theta$

$1=|2\overrightarrow{u}\times 3\overrightarrow{v}|$
$=6|\overrightarrow{u}\times \overrightarrow{v}|=6\sin \theta$
$\Rightarrow \sin \theta =\frac{1}{6}$ only one acute angle
$\therefore \theta ={\sin }^{-1}\left(\frac{1}{6}\right)$